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Questions and Answers
What are the objectives of learning regarding exponents and logarithms?
What are the objectives of learning regarding exponents and logarithms?
Evaluate ax in various cases, simplify algebraic expressions using exponent rules, evaluate logarithms, and solve equations using logarithm rules.
In the expression M = bn, what is referred to as the exponent?
In the expression M = bn, what is referred to as the exponent?
n
According to Rule 1 of exponents, what is the result of am × an?
According to Rule 1 of exponents, what is the result of am × an?
am+n
What is the value of 20?
What is the value of 20?
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What should b be in a logarithm to ensure M is positive?
What should b be in a logarithm to ensure M is positive?
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According to Rule 4 of logarithms, what is loga a equal to?
According to Rule 4 of logarithms, what is loga a equal to?
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The general form of logarithms is log b M = ___
The general form of logarithms is log b M = ___
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What is log2 1 equal to?
What is log2 1 equal to?
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What is the answer to 2^x = 16?
What is the answer to 2^x = 16?
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What is the solution to the logarithmic equation log2 40 + 3 log2 6 - log2 135?
What is the solution to the logarithmic equation log2 40 + 3 log2 6 - log2 135?
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What is the answer to the equation 25(10)^(2t) = 208?
What is the answer to the equation 25(10)^(2t) = 208?
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Study Notes
Exponents
- Exponential form is represented as ( M = b^n ) where ( b ) is the base and ( n ) is the exponent.
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Rules of Exponents:
- Rule 1: ( a^m \times a^n = a^{m+n} )
- Rule 2: ( \frac{a^m}{a^n} = a^{m-n} )
- Rule 3: ( (a^m)^n = a^{mn} )
- Rule 4: ( (ab)^m = a^m \times b^m )
- Rule 5: ( \sqrt[n]{a^m} = a^{\frac{m}{n}} )
- Rule 6: ( a^0 = 1 )
- Rule 7: ( a^{-m} = \frac{1}{a^m} )
Simplifying Expressions
- Examples to simplify:
- ( \frac{x^4 \times x^4}{x^2} ) results in ( x^{6} )
- Result of ( (x^{-1} y^{3})^{2} = \frac{y^{6}}{x^{2}} )
Solving Exponential Equations
- Solve ( 2^x = 16 ) leading to ( x = 4 )
- Solve ( \frac{a^3}{2} = 25 ) and find ( a )
Logarithms
- General form: ( \log_b M = n ) indicates ( b^n = M ).
- Logarithmic functions are inverses of exponential functions.
- Common logarithms (base 10) often omit the base; other bases should be indicated.
Rules of Logarithms
- Rule 1: ( \log_b (a \times c) = \log_b a + \log_b c )
- Rule 2: ( \log_b \left( \frac{a}{c} \right) = \log_b a - \log_b c )
- Rule 3: ( \log_b (a^n) = n \log_b a )
- Rule 4: ( \log_a a = 1 )
- Rule 5: ( \log_a 1 = 0 )
- Rule 6: ( b^{\log_b r} = r )
- Rule 7: ( \log_m b = \frac{\log_n b}{\log_n m} )
Example Problems with Logarithms
- Simplify ( \log_2 40 + 3 \log_2 6 - \log_2 135 ) resulting in ( 6 ).
- Solve ( 2^x = 7 ) and find ( x \approx 2.807 ).
- Solve ( 25^{x-1} = 3^x ) leading to ( x \approx 0.293 ).
Additional Practice
- Additional problems include solving equations of the form:
- ( 25(10)^{2t} = 208 )
- ( 38 + 12e^{-0.5t} = 208 )
- ( \log_{10} x + \log_{10} (x+9) = 1 )
- Aim to practice simplifying without calculators for improved skills.
Motivational Quote
- "Don't say you don't have enough time. You have the same number of hours per day as historical figures who achieved greatness."
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Description
Test your understanding of exponents and logarithms with this quiz. Explore the key rules and properties that govern exponential notation and logarithmic functions. Focus on the relationship between exponents and logarithms to enhance your mathematical skills.
